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\title{《基础复分析》第2章点集拓扑基础 - 习题}
\author{CGZ ET AL}

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% 《基础复分析》习题二

\begin{enumerate}

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\item  % 1 

设 $(X, d)$ 为度量空间。证明 $(X, \delta)$ 为有界的度量空间，其中
    $$
    \delta(x, y) = \frac{d(x, y)}{1 + d(x, y)}.
    $$
    
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\item  % 2 

设 $d_1(x, y)$ 与 $d_2(x, y)$ 为空间 $X$ 上的两个度量。
证明由它们定义的拓扑相容，当且仅当任给 $x \in X$ 以及 $\varepsilon > 0$, 存在 $\delta > 0$, 使得 $d_1(x, y) < \delta$ 蕴涵 $d_2(x, y) < \varepsilon$; 反之亦然。
    
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\item  % 3 

证明复平面在欧氏度量下定义的拓扑与在球面度量下诱导的拓扑相容。
    
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\item  % 4 

证明度量空间内任意子集的聚点组成一个闭集。
    
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\item  % 5 

令 
$$
A = \{(x, y) \in \mathbb{R}^2 : x = 0, |y| \leq 1\}
$$

$$
B = \{(x, y) \in \mathbb{R}^2 : x > 0, y = \sin(1/x)\}
$$

证明 $A \cup B$ 是连通的。
    
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\item  % 6 

设 $E \subsetneq \mathbb{C}$ 非空。证明 $\partial E$ 非空。
    
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\item  % 7 

证明可分度量空间中的一个集合是可数的，如果集合中的所有点都是孤立的。
    
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\item  % 8 

设 $E_1 \supseteq E_2 \supseteq \cdots$ 是 Hausdorff 空间中的非空紧致集序列，证明它们的交集非空。并举例说明如果 $E_i$ 只是闭的，则结论不一定成立。
    
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\item  % 9 

设 $X$ 是由所有实数序列 $\{x_n\}$ 组成的集合，使得每个序列只有有限项不为零。令 
$$
d(\{x_n\}, \{y_n\}) = \max |x_n - y_n|
$$

证明 $(X, d)$ 为不完备的度量空间。
    
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\item  % 10 

证明扩充复平面在球面度量下是紧致的。
    
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\item  % 11 

设 $X$ 与 $Y$ 是一个完备度量空间中的紧致集。
证明存在 $x \in X$, $y \in Y$ 使得 $d(x, y)$ 达到最小值。
    
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\item  % 12 

证明由开集定义的连续映射与由度量定义的连续映射相容。
    
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\item  % 13 

试构造一个将单位圆盘 $|z| < 1$ 映为整个平面的同胚。

\end{enumerate}

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